Computer Vision Exercise Session 1 Institute of Visual Computing - PowerPoint PPT Presentation

Computer Vision Exercise Session 1 Institute of Visual Computing

Organization  Teaching assistant  Bastien Jacquet  CAB G81.2  bastien.jacquet@inf.ethz.ch  Federico Camposeco  CNB D102.2  fede@inf.ethz.ch  Lecture webpage  http://www.cvg.ethz.ch/teaching/compvis/index.php Institute of Visual Computing

Organization  Assignments will be part of the final grade  Assignment every week (or every two weeks the second part)  1 or 2 weeks time to solve the assignment  Hand-in Thursday 13:00 (sharp!)  Bonus points can be used to compensate for missing points  MATLAB Download (www.stud-ides.ethz.ch) Institute of Visual Computing

Literature  Computer Vision: Algorithms and Application by Richard Szeliski, available online ( http://szeliski.org/Book/ )  Multiple View Geometry by Richard Hartley and Andrew Zisserman  Course Notes  http://cvg.ethz.ch/teaching/compvis/tutorial.pdf Institute of Visual Computing

Camera Calibration  Intrinsic parameters  K  Radial distortion coefficients 2D points 3D points  x PX   X  | x K R t Institute of Visual Computing

Camera Calibration  Use your own camera  Build your own calibration object  Print checkerboard patterns  Stich to two orthogonal planes Z Y X Institute of Visual Computing

Camera Calibration  4 Tasks:  Data normalization  Direct Linear Transform (DLT)  Gold Standard algorithm  Bouguet‘s Calibration Toolbox  Use the same settings for all tasks!  Good reference: Multiple View Geometry in computer vision ( Richard Hartley & Andrew Zisserman) Institute of Visual Computing

Data normalization  Shift the centroid of the points to the origin  Scale the points so that average distance to the origin is and , respectively. 3 2   Determine 1   s c ˆ 2 D x P    T s c   using normalized points. 2 D y   1    Determine ˆ   1  P T P U 1   s c 3 D x   s c    3 D y U   s c 3 D z   1   Institute of Visual Computing

Direct Linear Transform (DLT)   1 P     T  T T 0 w X x X   2   i i i i   0 AP P T T  T   0   w X y X 3 i i i i P     P  1 , 1    P     1 , 2   1 0 0 0 0 X X X x X x X x X x   ix iy iz i ix i iy i iz i           0 0 0 0 1 X X X y X y X y X y   ix iy iz i ix i iy i iz i   P 3 , 3   P   3 , 4 Institute of Visual Computing

Direct Linear Transform (DLT)  Singular Value Decomposition 2n 12 12 12 V‘ 12 2n 2n 2n A = U S Institute of Visual Computing

Direct Linear Transform (DLT)  Singular Value Decomposition 2n 12 12 12 V‘ 12 2n 2n 2n A = U S   P   1 , 1   P P P P   P 1 , 1 1 , 2 1 , 3 1 , 4 1 , 2     = P P P P      2 , 1 2 , 2 2 , 3 2 , 4     P P P P P   3 , 1 3 , 2 3 , 3 3 , 4 3 , 3   P   3 , 4 Institute of Visual Computing

Camera Matrix Decomposition (K and R)        | | P K R t KR KRC  K is upper triangular  R is orthonormal  QR decomposition A = QR  Q is orthogonal  R is upper triangular Institute of Visual Computing

Camera Matrix Decomposition (K and R)        | | P K R t KR KRC  M KR     1 1 1 M R K  Run QR decomposition on the inverse of the left 3x3 part of P  Invert both result matrices to get K and R Institute of Visual Computing

Camera Matrix Decomposition (C)  The camera center is the point for which  0 PC  This is the right null vector of P (  SVD) Institute of Visual Computing

Gold Standard Algorithm  Normalize data  Run DLT to get initial values  Compute optimal by minimizing the sum of ˆ P squared reprojection errors N min  ˆ ˆ 2 ˆ d( ) x , P X i i ˆ P  i 1  Denormalize ˆ P Institute of Visual Computing

Minimization in MATLAB  Fminsearch(...)  See code framework  Lsqnonlin(...)  nonlinear least-squares  Vectorize your parameters Institute of Visual Computing

Bouguet‘s Calibration Toolbox  Download and install the toolbox: (http://www.vision.caltech.edu/bouguetj/calib_doc/index.html)  Go through the tutorial and learn how to calibrate a camera with that toolbox  Print your own calibration pattern (available on the website)  Use the toolbox for calibration and compare the result with the results of your own calibration algorithm Institute of Visual Computing

Hand-in  Source code  Matlab .mat file with hand-clicked 3D-2D correspondences  Image used for calibration  Visualize hand-clicked points and reprojected 3D points  Discuss and compare values of calibration obtained for all methods  Discuss average reprojection error of all methods. Institute of Visual Computing

Some hints  Work with normalized homogeneous coordinates always. Camera calibration K should respect this convention.  Check that the obtained orientation R correspond to the expected world value, otherwise K will have negative values in its diagonal.  When reprojecting points use average of reprojection error for comparison and remember to normalize reprojected coordinates to w = 1.  Remember to use the same camera with the same settings for all tasks! Institute of Visual Computing

Hand-in  Example reprojection of the the 3D points Institute of Visual Computing